On Sun, Mar 23, 2008 at 03:37:27AM -0700, Brian Tenneson wrote:
> 
> <begin their argument for the non-existence for the universe>
> Definition: To contain means <insert something most people would
> accept here>.  The notation and word for 'is contained in' is
> is<in.
> 
> Thing and exists are undefined or ... acceptably defined only be
> common intuitive sense of what a thing is, but neither formally (in
> her argument)
> 
> Definition: the universe (call it U) is a thing that has the property
> that it contains all things, notated by (x) (x is<in U), where x is a
> thing.
> 
> Theorem: If the universe exists then the (three or so) axioms of
> binary logic are inconsistent.
> Proof: The method is to show that if U exists then there is a logical
> statement (ie, a WELL FORMED formula) that is true if and only it is
> false, being simultaneously, to abuse language, true and not true,
> which violates the +definition+ of the words not and and.
> 
> Suppose U exists.  Then apply Russell's approach.  Given how broad and
> vague 'thing' is defined, let's discuss the thing, call it S, this
> thing called S is the thing that contains all things that don't
> contain themselves.  In the notation, let S be the thing (given the
> vagueness of 'thing', S is a thing) such that
> (x) (x is<in S if and only if x!<x).
> In other words, S is the thing such that for all things x, x is
> contained in S if and only if x is not contained in x.
> 
> Since we wrote (x), then apply to S by an application of some
> universal quantifier rule, which most people would accept (and maybe
> they should qualify the universal sometimes) to S. Then you get, just
> as Russell's approach:
> 
> (S is<inS if and only if S!<S).
> 
> This contradiction proves the theorem.  That if the universe exists,
> then binary logic is inconsistent.
> 
> 
> Corollary: The universe does not exist.
> "Proof:"  Binary logic is consistent, therefore, by contraposition of
> the theorem, the universe does not exist.
> 
> <end their argument for the non-existence for the universe>
> 
> 
> I've been banging away at this keyboard for a while so I'll post this
> and take a break.
> 
> The idea came to me when I tried basically to prove her argument that
> the universe does NOT exist, wrong.  It occurred to me that three
> truth values are sufficient to make the usual proof by contradiction
> +not a tautology+.  And, therefore, even in 3-valued logic, her
> argument fails.
> 
> Obviously, that doesn't prove the universe does exist, it just proves
> her argument that is doesn't is wrong.
> 
> 
> 
> 
> 
> 
> 
> <end their argument for the non-existence for the universe>
> 

Maybe it instead proves that "things" like S do not exist in the
universe. OK, it means we have to change the definition of universe a
bit, but this is not so strange as universe really just means all that exists.

So, yeah, I'd say it was a bit of linguistic sophistry, rather than
being too profound.

Anyway, the question of whether Russell's paradox can be found to not
hold force in non-standard logic seems interesting, and potentially
well motivated for the MUH case which ab initio would include things
like S in the level 4 "multiverse".

Cheers

----------------------------------------------------------------------------
A/Prof Russell Standish                  Phone 0425 253119 (mobile)
Mathematics                              
UNSW SYDNEY 2052                         [EMAIL PROTECTED]
Australia                                http://www.hpcoders.com.au
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